Logic arrangement, data structure, system and method for multilinear representation of multimodal data ensembles for synthesis, recognition and compression

ABSTRACT

A data structure, method, storage medium and logic arrangement are provided for use in collecting and analyzing multilinear data describing various characteristics of different objects. In particular it is possible to recognize an unknown individual, an unknown object, an unknown action being performed by an individual, an unknown expression being formed by an individual, as well as synthesize a known action never before recorded as being performed by an individual, synthesize an expression never before recorded as being formed by an individual, and reduce the amount of stored data describing an object or action by using dimensionality reduction techniques, and the like.

CROSS REFERENCE TO RELATED APPLICATIONS

The present application claims priority from U.S. Patent Application Ser. Nos. 60/337,912, 60/383,300 and 60/402,374, the entire disclosures of which are incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates generally to a logic arrangement, data structure, system and method for acquiring data, and more particularly to a logic arrangement, data structure, system and method for acquiring data describing at least one characteristic of an object, synthesizing new data, recognizing acquired data and reducing the amount of data describing one or more characteristics of the object (e.g., a human being).

BACKGROUND OF THE INVENTION

Natural images are the composite consequence of multiple factors related to scene structure, illumination and imaging. Human perception of natural images remains robust despite significant variation of these factors. For example, people possess a remarkable ability to recognize faces given a broad variety of facial geometries, expressions, head poses and lighting conditions.

Some past facial recognition systems have been developed with the aid of linear models such as principal component analysis (“PCA”), independent component analysis (“ICA”). Principal components analysis (“PCA”) is a popular linear technique that has been used in past facial image recognition systems and processes. By their very nature, linear models work best when a single-factor varies in an image formation. Thus, linear techniques for facial recognition systems perform adequately when person identity is the only factor permitted to change. However, if other factors (such as lighting, viewpoint, and expression) are also permitted to modify facial images, the recognition rate of linear facial recognition systems can fall dramatically.

Similarly, human motion is the composite consequence of multiple elements, including the action performed and a motion signature that captures the distinctive pattern of movement of a particular individual. Human recognition of particular characteristics of such movement can be robust even when these factors greatly vary. In the 1960's, the psychologist Gunnar Kohansson performed a series of experiments in which lights were attached to people's limbs, and recorded a video of the people performing different activities (e.g., walking, running and dancing). Observers of these moving light videos in which only the lights are visible were asked to classify the activity performed, and to note certain characteristics of the movements, such as a limp or an energetic/tired walk. It was observed that this task can be performed with ease, and that the observer could sometimes determine even recognize specific individuals in this manner. This may corroborate the idea that the motion signature is a perceptible element of human motion. and that the signature of a motion is a tangible quantity that can be separated from the actual motion type.

However, there is a need to overcome at least some of the deficiencies of the prior art techniques.

OBJECTS AND SUMMARY OF THE INVENTION

Such need is addressed by the present invention. One of the objects of the present invention is to provide a logic arrangement, data structure, storage medium, system and method for generating an object descriptor. According to an exemplary embodiment of the present invention such data structure can include a plurality of first data elements that have information regarding at least one characteristic of the at least one object. The information of the first data elements is capable of being used to obtain the object descriptor. The object descriptor is related to the at least one characteristic and a further characteristic of the at least one object, and is capable of being used to generate a plurality of second data elements which contain information regarding the further characteristic of the at least one object based on the object descriptor.

In another exemplary embodiment of the present invention, the method can include a plurality of first data elements containing information regarding at least one characteristic of the at least one object. The object descriptor is obtained based on the information of the first data elements and is related to the at least one characteristic and a further characteristic of the object. A plurality of second data elements containing information regarding the further characteristic of the at least one object based on the object descriptor.

In still another exemplary embodiment of the present invention, the storage medium including a software program, which when executed by a processing arrangement, is configured to cause the processing arrangement to execute a series of steps. The series of steps can include a plurality of first data elements containing information regarding at least one characteristic of the at least one object. The object descriptor is obtained based on the information of the first data elements and is related to the at least one characteristic and a further characteristic of the object. A plurality of second data elements containing information regarding the further characteristic of the at least one object based on the object descriptor.

In a further exemplary embodiment of the present invention, the logic arrangement is adapted for an execution by a processing arrangement to perform a series of steps. The series of steps can include a plurality of first data elements containing information regarding at least one characteristic of the at least one object. The object descriptor is obtained based on the information of the first data elements and is related to the at least one characteristic and a further characteristic of the object. A plurality of second data elements containing information regarding the further characteristic of the at least one object based on the object descriptor.

Another of the objects of the present invention is to provide a logic arrangement, data structure, storage medium, system and method for identifying a sample object of a plurality of objects based upon a sample object descriptor. According to an exemplary embodiment of the present invention such data structure can include a plurality of first data elements that have information which is defined by at least two first primitives. The first data elements are capable of being used to obtain at least one of a plurality of object descriptors. The exemplary data structure may also include a plurality of second data elements that have information which is defined by at least two second primitives. The second data elements are capable of being used to obtain the sample object descriptor. The at least one obtained object descriptor configured to be compared to the sample object descriptor for determining whether the object is potentially identifiable as one of the object descriptors. Each of the plurality of object descriptors is associated with a respective one of a plurality of objects.

In another exemplary embodiment of the present invention, the method can include a plurality of data elements which are defined by at least two primitives are collected. At least one of a plurality of object descriptors are obtained based on the information of the data elements. The sample object descriptor is compared to at least one of the object descriptors for determining whether the sample object descriptor is identifiable as one of the object descriptors. Each of the object descriptors is associated with a respective one of a plurality of objects.

In still another exemplary embodiment of the present invention, the storage medium including a software program, which when executed by a processing arrangement, is configured to cause the processing arrangement to execute a series of steps. The series of steps can include can include a plurality of data elements which are defined by at least two primitives are collected. At least one of a plurality of object descriptors are obtained based on the information of the data elements. The sample object descriptor is compared to at least one of the object descriptors for determining whether the sample object descriptor is identifiable as one of the object descriptors. Each of the object descriptors is associated with a respective one of a plurality of objects.

In a further exemplary embodiment of the present invention, the logic arrangement is adapted for an execution by a processing arrangement to perform a series of steps. The series of steps can include a plurality of data elements which are defined by at least two primitives are collected. At least one of a plurality of object descriptors are obtained based on the information of the data elements. The sample object descriptor is compared to at least one of the object descriptors for determining whether the sample object descriptor is identifiable as one of the object descriptors. Each of the object descriptors is associated with a respective one of a plurality of objects.

Yet another of the objects of the present invention is to provide a logic arrangement, data structure, storage medium, system and method for reducing the dimensionality of one of the at least two object descriptors. According to an exemplary embodiment of the present invention such data structure can include a plurality of data elements that have information defined by at least two primitives. The data elements are capable of being used to obtain one of the object descriptors. The one of the object descriptors is capable having a reduced dimensionality.

In another exemplary embodiment of the present invention, the method can include a plurality of data elements defined by at least two primitives are collected. The one of the object descriptors based on the information of the data elements is obtained. The dimensionality of the one of the object descriptors is reduced.

In still another exemplary embodiment of the present invention, the storage medium including a software program, which when executed by a processing arrangement, is configured to cause the processing arrangement to execute a series of steps. The series of steps can include can include a plurality of data elements defined by at least two primitives are collected. The one of the object descriptors based on the information of the data elements is obtained. The dimensionality of the one of the object descriptors is reduced.

In a further exemplary embodiment of the present invention, the logic arrangement is adapted for an execution by a processing arrangement to perform a series of steps. The series of steps can include a plurality of data elements defined by at least two primitives are collected. The one of the object descriptors based on the information of the data elements is obtained. The dimensionality of the one of the object descriptors is reduced.

BRIEF DESCRIPTION OF THE DRAWINGS

Further objects, features and advantages of the invention will become apparent from the following detailed description taken in conjunction with the accompanying figures showing illustrative embodiments of the invention, in which:

FIG. 1 is a block diagram of a data analysis system according to an exemplary embodiment of the present invention;

FIG. 2 is a flow diagram of an exemplary embodiment of a process according to the present invention which analyzes multilinear data;

FIG. 3 is a flow diagram of an exemplary embodiment of a core tensor computation procedure of the process of FIG. 2 which performs an N-mode SVD algorithm for decomposing an N-dimensional tensor;

FIG. 4 is a flow diagram of an exemplary embodiment of a process of FIG. 2 which synthesizes the remaining actions for a new individual;

FIG. 5 is a flow diagram of an exemplary embodiment of an action generation procedure of the process of FIG. 2 which synthesizes an observed action for a set of individuals;

FIG. 6 is a flow diagram of an exemplary embodiment of an individual recognition procedure of the process of FIG. 2 which recognizes an unidentified individual performing a known actions as one of a group of individuals;

FIG. 7 is a flow diagram of an exemplary embodiment of an action recognition procedure of the process of FIG. 2 which recognizes an unknown action being performed by a known individual;

FIG. 8 is a flow diagram of another exemplary embodiment of a process according to the present invention which analyzes multilinear data;

FIG. 9 is a flow diagram of an exemplary embodiment of the individual recognition procedure of the process of FIG. 8 which recognizes an unidentified individual given an unknown facial image;

FIG. 10 is a flow diagram of an exemplary embodiment of the expression recognition procedure of the process of FIG. 8 which recognizes of an unidentified expression being displayed by a known person;

FIG. 11 is a flow diagram of an exemplary embodiment of a data reduction process of the process of FIG. 8 which dimensionally reduces the amount of data describing an individual displaying an expression;

FIGS. 12A-12F are block diagrams of sample tensors and equivalent mode-1, mode-2 and mode-3 flattened tensors according to an exemplary embodiment of the present invention;

FIG. 13 is a flow diagram of another exemplary embodiment of a process according to the present invention which analyzes multilinear data;

FIG. 14 is a flow diagram of an exemplary embodiment of a core matrix computation procedure of the process of FIG. 13 which performs an SVD matrix algorithm for decomposing a matrix; and

FIG. 15 is a flow diagram of an exemplary embodiment of a process of FIG. 13 which synthesizes the remaining actions for a new individual.

Throughout the figures, the same reference numerals and characters, unless otherwise stated, are used to denote like features, elements, components or portions of the illustrated embodiments. Moreover, while the present invention will now be described in detail with reference to the figures, it is done so in connection with the illustrative embodiments. It is intended that changes and modifications can be made to the described embodiments without departing from the true scope and spirit of the subject invention as defined by the appended claims.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

FIG. 1 illustrates an exemplary embodiment of a data analysis system 100 for use in the collection and analysis of data describing various characteristics of different objects. In this embodiment, a central server 102 is provided in the system 100, which provides therein a central processing unit (“CPU”) 104, a data storage unit 106 and a database 108. The central server 102 is connected to a communications network 110, which is in turn connected to an data capturing system 112. The data capturing system 112 can include at least one camera (not shown for the sake of clarity). A first client server 114 is provided in the system 100, which provides therein a CPU 116, a data storage unit 118, and a database 120. The first client server 114 is connected to the communications network 110. A second client server 124 is also provided in the system 100, which situates a CPU 126, a data storage unit 128, and a database 130. The second client server 124 is also connected to the communications network 110. It should be understood that the central server 102, the image capture system 112, the first client server 114 and the second client server 124 can forward data messages to each other over the communications network 110.

In a preferred embodiment of the present invention, the data capturing System 112 can be a “VICON” system which employs at least four video cameras. The VICON system can be used to capture human limb motion and the like.

A multilinear data analysis application can be stored in the data storage unit 106 of the central server 102. This multilinear data analysis application is capable of recognizing an unknown individual, an unknown object, an unknown action being performed by an individual, an unknown expression, an unknown illumination, an unknown viewpoint, and the like. Such application can also synthesize a known action that has never before recorded as being performed by an individual, as well as an expression which has previously not been recorded as being formed by an individual. Further the application can reduce the amount of stored data that describes an object or action by using dimensionality reduction techniques, and the like. It should be understood that dimensionality reduction is equivalent to compression and data reduction. The multilinear data analysis application preferably utilizes a corpus of data, which is collected using the data capturing system 112 from different subjects. The corpus of data is stored in the database 108 of the server 102, and can be organized as a tensor D, which shall be described in further detail as follows.

A tensor, also known as an n-way array or multidimensional matrix or n-mode matrix, is a higher order generalization of a vector (first order tensor) and a matrix (second order tensor). A tensor can be defined as a multi-linear mapping over a set of vector spaces. The tensor can be represented in the following manner: A∈IR^(I) ¹ ^(×I) ² ^(× . . . ×I) ^(N) where A is a tensor. The order of the tensor A is N. A tensor is formed by a group of primatives. Each primative is a set of mode vectors, such that a first primative is a set of mode-1 vectors, a second vector is a set of mode-2 vectors, an n^(th) primative is a set of mode-n vectors, etc. In an alternate embodiment, the primatives can be row vectors of a matrix, column vectors of a matrix, index of a vector, etc. An element of tensor A is denoted as A_(i) ₁ _(. . . i) _(n) _(. . . i) _(N) or a_(i) ₁ _(. . . i) _(n) _(. . . i) _(N) or where 1≦i_(n≦I) _(n). Scalars are denoted by lower case letters (a, b, . . . ), vectors by bold lower case letters (a, b . . . ), matrices by bold upper-case letters (A, B . . . ), and higher-order tensors by italicized bolded upper-case letters (A, B . . . ).

In tensor terminology, column vectors are referred to as mode-1 vectors, and row vectors are referred to as mode-2 vectors. Mode-n vectors of an N^(th) order tensor A∈IR^(I) ¹ ^(×I) ² ^(× . . . ×I) ^(N) are the I_(n)-dimensional vectors obtained from the tensor A by varying index i_(n) while maintaining the other indices as fixed. The mode-n vectors are the column vectors of matrix A_((n))∈IR^(I) ^(n) ^(×(I) ¹ ^(I) ² ^(. . . I) ^(n−1) ^(I) ^(n+1) ^(. . . I) ^(N) ⁾ that can result from flattening the tensor A, as shown in FIGS. 12A-12F. The flattening procedure shall be described in further detail below. The n-rank of tensor A∈ IR^(I) ¹ ^(×I) ² ^(× . . . ×I) ^(N) , is defined as the dimension of the vector space generated by the mode-n vectors:

R _(n)=rank_(n)(A)=rank(A _(n))).

FIGS. 12A-12C show third order tensors 1200, 1210, 1220, respectively, each having dimensions I_(I)×I₂×I₃. FIG. 12D shows the third order tensor 1200 after having been mode-1 flattened to obtain a matrix 1250 containing mode-1 vectors of the third order tensor 1200. The third order tensor 1200 of FIG. 12A is a cube type structure, while the matrix 1250 is a two dimensional type structure having one index, i.e., I₂, imbedded (to a certain degree) within the matrix 1250. FIG. 12E shows a matrix 1260 containing mode-2 vectors of the third order tensor 1210 after it has been mode-2 flattened. This third order tensor 1210 is a cube type structure, while the matrix 1260 is a two dimensional type structure having one index, e.g., I₃, imbedded (to a certain degree) with the data. FIG. 12F shows the third order tensor 1220 after having been mode-3 flattened to obtain a matrix 1270 containing mode-3 vectors of the third order tensor 1220. Such third order tensor 1220 is a cube type structure, while the matrix 1270 organizes is a two dimensional type structure having one index, e.g., I₁, imbedded (to a certain degree) with the data.

A generalization of the product of two matrices can be the product of the tensor and matrix. The mode-n product of tensor A∈IR^(I) ¹ ^(×I) ² ^(× . . . ×I) ^(n) ^(× . . . I) ^(N) by a matrix M∈IR^(J) ^(n) ^(×I) ^(n) , denoted by A×_(n)M, is a tensor B∈IR^(I) ¹ ^(× . . . ×I) ^(n−1) ^(×J) ^(n) ^(×I) ^(n) ^(× . . . ×I) ^(N) , whose entries are

$B_{i_{{1\mspace{14mu} \ldots}\mspace{14mu}}i_{n - 1}j_{n}i_{{n + {1\mspace{14mu} \ldots}}\mspace{14mu}}i_{N}} = {\sum\limits_{i_{n}}{a_{i_{1}}\ldots \mspace{14mu} i_{n - 1}i_{n + 1}\mspace{14mu} \ldots \mspace{14mu} i_{N}^{m}j_{n}{i_{n}.}}}$

The entries of the tensor B are computed by

$\left( {A \times_{n}M} \right)_{i_{1}\mspace{14mu} \ldots \mspace{14mu} i_{n - 1}j_{n}i_{n + 1}\mspace{14mu} \ldots \mspace{14mu} i_{N}} = {\sum\limits_{i_{n}}{a_{i_{1}\mspace{14mu} \ldots \mspace{14mu} i_{n - 1}i_{n}i_{n + 1}\mspace{14mu} \ldots \mspace{14mu} i_{N}}{m_{j_{n}i_{n}}.}}}$

The mode-n product can be expressed as B=A×_(n)M, or in terms of flattened matrices as B_(n)=MA_((n)). The mode-n product of a tensor and a matrix is a special case of the inner product in multilinear algebra and tensor analysis. The mode-n product is often denoted using Einstein summation notation, but for purposes of clarity, the mode-n product symbol will be used. The mode-n product has the following properties:

-   1. Given a tensor A∈IR^(I) ¹ ^(× . . . ×I) ^(n) ^(× . . . I) ^(m)     ^(. . .) and two matrices, U∈IR^(J) ^(m) ^(×I) ^(m) and V∈IR^(J)     ^(n) ^(×I) ^(n) the following property holds true:

$\begin{matrix} {{A \times_{m}U \times_{n}V} = {\left( {A \times_{m}U} \right) \times_{n}V}} \\ {= {\left( {A \times_{n}V} \right) \times_{m}U}} \\ {= {A \times_{n}V \times_{m}U}} \end{matrix}$

-   2. Given a tensor A∈IR^(I) ¹ ^(× . . . ×I) ^(n) ^(× . . . I) ^(N)     and two matrices, U∈IR^(J) ^(n) ^(×I) ^(n) and V∈IR^(K) ^(n) ^(×J)     ^(n) the following property holds true:

(A× _(n) U)×_(n) V=A× _(n)(VU)

An N^(th) order tensor A∈IR^(I) ¹ ^(×I) ² ^(× . . . ×I) ^(N) has a rank-1 when it is able to be expressed as the outer product of N vectors: A=u₁∘u₂∘ . . . ∘u_(N). The tensor element is expressed as α_(ij . . . m)=u_(1i)u_(2j) . . . u_(Nm), where u_(1i) is the i^(th) component of u₁, etc. The rank of a N^(th) order tensor A, denoted R=rank(A), is the minimal number of rank-1 tensors that yield A in a linear combination:

$A = {\sum\limits_{r\; = 1}^{R}{\sigma_{r}{u_{1}^{(r)} \cdot u_{2}^{(r)} \cdot \ldots \cdot {u_{N}^{(r)}.}}}}$

A singular value decomposition (SVD) can be expressed as a rank decomposition as is shown in the following simple example:

$\begin{matrix} {M = {{\begin{bmatrix} a & b \\ c & d \end{bmatrix}\begin{bmatrix} \sigma_{11} & 0 \\ 0 & \sigma_{22} \end{bmatrix}}\begin{bmatrix} f & g \\ h & i \end{bmatrix}}} \\ {= {{{\sigma_{11}\begin{bmatrix} a \\ c \end{bmatrix}} \cdot \begin{bmatrix} f \\ g \end{bmatrix}} + {{\sigma_{22}\begin{bmatrix} b \\ d \end{bmatrix}} \cdot}}} \\ {= {U_{1}\Sigma \; U_{2}^{T}}} \\ {= {{\left\lbrack {u_{1}^{(1)}u_{1}^{(2)}} \right\rbrack \begin{bmatrix} \sigma_{11} & 0 \\ 0 & \sigma_{22} \end{bmatrix}}\left\lbrack {u_{2}^{(1)}u_{2}^{(2)}} \right\rbrack}^{T}} \\ {= {\sum\limits_{i = 1}^{R = 2}{\sum\limits_{j = 1}^{R = 2}{\sigma_{ij}{u_{1}^{(i)} \cdot u_{2}^{(j)}}}}}} \end{matrix}$

It should be noted that an SVD is a combinatorial orthogonal rank decomposition, but that the reverse is not true; in general, rank decomposition is not necessarily singular value decomposition. Also, the N-mode SVD can be expressed as an expansion of mutually orthogonal rank-1 tensors, as follows:

${D = {\sum\limits_{i_{1} = 1}^{R_{1}}\mspace{14mu} {\ldots \mspace{14mu} {\sum\limits_{i_{n} = 1}^{R_{n}}\mspace{14mu} {\ldots \mspace{14mu} {\sum\limits_{i_{N} = 1}^{R_{N}}\mspace{14mu} {\ldots \mspace{14mu} Z_{i_{1}}\mspace{11mu} {\ldots \mspace{14mu}}_{i_{N}}{U_{1}^{(i_{t})} \cdot \ldots \cdot U_{n}^{({i\; n})} \cdot \ldots}\mspace{14mu} U_{N}^{({iN})}}}}}}}},$

where U_(n) ^((in)) is the i_(n) column vector of the matrix U_(n). This is analogous to the equation

$\sum\limits_{i = 1}^{R = 2}{\sum\limits_{j = 1}^{R = 2}{\sigma_{ij}{u_{1}^{(i)} \cdot {u_{2}^{(j)}.}}}}$

A client interface application can be stored in the data storage units 118, 128 of the first and second client servers 114, 124, respectively. The client interface application preferably allows the user to control the multilinear data analysis application described previously. For example, the client interface application can instruct the multilinear data analysis application to generate new data describing a particular characteristic of a known object that may be different from those characteristics of the known object which were already observed. In addition, the client interface application can instruct the multilinear data analysis application to generate new data describing a particular characteristic of the remainder of the population of observed objects that are different from those characteristics of the remainder of the population already observed. Also, the client interface application can instruct the multilinear data analysis application to recognize an unknown object from the population of observed objects, recognize a characteristic of a known object from the characteristics of the known object already observed, dimensionally reduce the amount of data stored to describe a characteristic of a known object, etc. In one exemplary embodiment of the present invention, the object can be a person and the characteristic may be an action. In another embodiment of the present invention, the object could be a person's face, and the characteristic can be a facial expression. In response to the client interface application's instructions, the multilinear data analysis application may transmit to the client interface application certain information describing the requested characteristic or object.

A. Motion Signature Using A Tensor Representation of A Corpus of Data

FIG. 2 illustrates flow diagram of an exemplary embodiment of a process 200 which is indicative of the multilinear data analysis application. As described above for the multilinear data analysis application, the process 200 is configured to recognize the unknown subject or individual, recognize the unknown action being performed by the known subject, generate a known action never before recorded as being performed by the subject, etc. In particular the multilinear data analysis application utilizes the corpus of motion data, which is collected using the data capturing system 112 from different subjects. This corpus of motion information is stored in the database 108 of the server 102, and describes angles of the joints in the legs of at least one subject performing at least one action. The corpus of motion information can be organized as a tensor D. It should be understood that the corpus of motion information can also be organized as a matrix D or a vector d. For example, if the information is organized as a matrix D, the process 200 preferably remains the same, but the underlying tensor procedures could be converted to matrix procedure equivalents. It should also be noted that representing the data contained in the tensor D may integrate multiple indices into a singular matrix index. Likewise, if the information is organized as a vector d, the process 200 preferably remains the same, but the underlying tensor procedures could be converted to vector procedure equivalents. It should also be noted that representing the data contained in the tensor D may integrate multiple indices into a singular vector index.

The corpus of motion data is preferably collected from different subjects that perform at least one action which forms the tensor D. Each action can be repeated multiple times, and a motion cycle can be segmented from each motion sequence. For example, in order to suppress noise, the collected motion data can be passed through a low-pass fourth-order Butterworth filter at a cut off frequency of 6 Hz, and missing data may be interpolated with a cubic spline. Joint angles can be computed to represent the motion information of the limbs of various subjects (e.g., people). To compute the joint angles, the frame coordinate transformation for each limb may be calculated with respect to an area in which the motion information is collected, the relative orientation of each limb in the kinematic chain can then be determined, and the inverse kinematic equations are thus obtained. The joint angles are thereafter stored in the tensor D. Such tensor D can have the form of a IR^(G×M×T), where G is the number of subjects, M is the number of action classes, and T is the number of joint angle time samples.

In an exemplary implementation of a preferred embodiment according to the present invention, three motions are collected for each person: e.g., walk, ascend-stairs, and descend stairs. In another exemplary implementation, each action can be repeated ten (10) times. In yet another exemplary implementation, human limb motion can be recorded using the VICON system that employs four infra-red video cameras. These cameras generally detect infra-red light which is reflected from 18 markers, 9 placed on each leg of a human subject. The system 112 then computes a three-dimensional position of the markers relative to a fixed coordinate frame. The video cameras can be positioned on one side of a 12 meter long walkway such that each marker can be observed by at least two cameras during the subject's motion. To extract the three angles spanned by a joint of the subject, a plane can be defined for each limb whose motion can be measured relative to the sagittal, frontal and transverse planes through the body of the subject. It should be noted that the joint angle time samples reflect the joint angles of various joints as they move over time.

Turning to further particulars of FIG. 2, in step 202, the process 200 collects motion information or data on various subjects (e.g., people) performing different actions, e.g., new motion data. The motion is collected as a group of vectors. Each of the group of vectors represents a subject performing an action. If each of the possible the actions and the individual are known, the data can be integrated into the tensor D. If the action or individual are not known, such data would likely not be integrated into the tensor D until those pieces of information are determined. The data describing an unknown action or individual is organized as a new data tensor D_(p,a) of a new subject or a new data vector d of a new subject. The new data tensor D_(p,a) includes more than one new data vector d. Each new data vector d of the new data tensor D_(p,a) describes the motion of subject p performing action a.

At step 204, the process 200 solves for a core tensor Z which can be generally used for defining the inter-relationships between the orthonormal mode matrices. This step represents an N-mode singular value decomposition (“SVD”) process 204, shown in FIG. 3, and described in further detail herein. It should be noted that the N-mode SVD procedure of step 204 is an orthonormal decomposition procedure. The N-mode SVD procedure of step 204 solves for the core tensor Z.

When this procedure of step 204 determines the core tensor Z, the process 200 advances to step 205.

In an alternate embodiment of the present invention, an alternate n-mode orthonormal decomposition procedure is used in place of the n-mode SVD procedure.

In step 205, the process 200 analyzes the data collected in the step 202. With the knowledge of motion sequences of several subjects, the tensor D can take the form of a IR^(G×M×T) tensor, where G is the number of subjects or people, M is the number of action classes, and T is the number of joint angle time samples. The N-mode SVD procedure of step 204 decomposes the tensor D into the product of a core tensor Z, and three orthogonal matrices as follows:

D=Z× ₁ P× ₂ A× ₃ J,

The subject matrix P=[p₁ . . . p_(n) . . . p_(G)]^(T), whose subject-specific row vectors p_(n) ^(T) span the space of person parameters, encodes the per-subject invariance across actions. Thus, the matrix P contains the subject or human motion signatures. The action matrix A=[a₁ a_(m) a_(m)]^(T), whose action specific row vectors a_(n) ^(T) span the space of action parameters, encodes the invariance for each action across different subjects. The joint angle matrix J whose row vectors which span the space of joint angles are preferably the eigenmotions, the motion variation.

The product Z×₃J transforms the eigenmotions into tensormotions, a tensor representation of the variation and co-variation of modes (subjects and action classes). The product Z×₃J also characterizes how the subject's parameters and action parameters interact with one another. The tensor

B=Z× ₂ A× ₃ J

is an action specific tensormotion, which contains a set of basis matrices for all the motions associated with particular actions. The tensor

C=Z× ₁ P× ₃ J

is a subject/signature specific tensormotion, which preferably contains a set of basis matrices for all the motions associated with particular subjects (with particular subject motion signatures). The core tensor Z, the matrix A, and the matrix J generated by the N-mode SVD procedure of step 204 of the tensor D define a generative model.

In step 206, the process 200 determines whether it has been instructed by the client interface application to synthesize new data describing at least one known action that was never before recorded as being performed by a new subject. If the process 200 has received such instruction, step 208 is executed to perform advances to an individual generation procedure, as shown in further detail in FIG. 4 and described herein. When the individual generation procedure of step 208 is complete, the process 200 advances to step 226.

In step 210, the process 200 determines if it was instructed by the client interface application to synthesize new data describing anew action that was never before recorded as being performed by the remainder of the population of observed subjects. If the process 200 has received such instruction, the process 200 continues to an action generation procedure of step 212, as shown in further detail in FIG. 5 and described herein. When the action generation procedure of step 212 is completed, the process 200 is forwarded to step 226.

In step 214, the process 200 determines if it was instructed by the client interface application to recognize an unknown subject who has been observed to perform a known action as one of the population of observed known subjects. If the process 200 has received such instruction, the process 200 is directed to an individual recognition procedure of step 216, as shown in greater detail in FIG. 6 and described infra. Once the individual recognition process 216 is completed, the process 200 advances to step 226.

In a preferred embodiment, the process 200 is capable of recognizing an unknown subject who has been observed performing an unknown action as one of the population of observed known subjects.

In step 218, the process 200 determines if it was instructed by client interface application to recognize an unknown action being performed by a known subject as one of the actions already observed as being performed by the known subject. If the process 200 has received such an instruction, the process 200 continues to an action recognition procedure of step 220, as shown in FIG. 7 and described infra. When the individual recognition procedure of step 220 is completed, the process 200 is forwarded to step 226. Then in step 226, the process 200 determines whether a data set for a new subject should be integrated into the tensor D or if the client interface application has transmitted a new instruction. In particular, if a data set for a new subject is available, the process 200 advances to step 202. Otherwise, the process 200 received the new instruction from the client interface application, so the process 200 continues to step 206.

FIG. 3 illustrates the exemplary details N-mode SVD procedure of step 204 for performing an N-mode SVD algorithm to decompose the tensor D and compute the core tensor Z. The N-mode SVD procedure of step 204 is related to and grows out of a natural generalization of the SVD procedure for a matrix. For example, a matrix D∈IR^(I) ¹ ^(×I) ² is a two-mode mathematical object that has two associated vector spaces, e.g., a row space and a column space. The SVD procedure for a matrix orthogonalizes these two spaces, and decomposes the matrix as D=U₁ΣU₂ ^(T) with the product of an orthogonal column-space represented by the left matrix U₁∈IR^(I) ¹ ^(×J) ¹ , a diagonal singular value matrix Σ∈IR^(J) ¹ ^(×J) ² , and an orthogonal row space represented by the right matrix U₂∈IR^(I) ² ^(×J) ² . In terms of the mode-n products defined above, this matrix product can be rewritten as D=Σ×₁U₁×₂U₂. If the data contained within the tensor D is represented as a matrix D, the SVD procedure for a matrix can be used.

By extension, the tensor D can be an order-N tensor comprising N spaces, where N is preferably greater than 2. N-mode SVD is a natural generalization of SVD that orthogonalizes these N spaces, and decomposes the tensor as the mode-n product of N-orthonormal spaces.

D=Z× ₁ U ₁×₂ U ₂ . . . ×_(n) U _(n) . . . ×_(N) U _(N),

A matrix representation of the N-mode SVD can be obtained by:

D _((n)) =U _(n) Z _((n))(U_(n+1)

U _(n+2)

. . .

U _(N)

U ₁

. . .

U _(n−1))^(T)

where

is the matrix Kronecker product. The core tensor Z, can be analogous to the diagonal singular value matrix in conventional matrix SVD. It is important to realize, however, that the core tensor does not have a diagonal structure; rather, Z is in general a full tensor. The core tensor Z governs the interaction between mode matrices U_(n), for n=1, . . . , N. Mode matrix U_(n) contains the orthonormal vectors spanning the column space of the matrix D_((n)) that results from the mode-n flattening of the tensor D, as illustrated in FIGS. 12A-12F.

As shown in FIG. 3, the procedure of step 204 begins at step 302 by setting an index n to one (1). This allows the process 204 to begin computing an initial matrix from the tensor D. When the index n is set to one, the procedure of step 204 advances to step 304. In step 304, the procedure of step 204 computes the matrix U_(n) as defined by D=Z×₁U₁X₂U₂ . . . ×_(n)U_(n) . . . ×_(N)U_(N), by computing the SVD of the flattened matrix D_((n)). Once the matrix U_(n) is computed, the procedure of step 204 continues to step 306. In step 306 the procedure of step 204 sets the matrix U_(n) to be a left matrix of the SVD. Once the matrix U_(n) is set appropriately, the procedure of step 204 goes on to step 308, in which it is determined whether the index n is equal to the order of the tensor, i.e. N. If the index n is equal to the order of the tensor, the procedure of step 204 advances to step 312. Otherwise, the process 204 is forwarded to step 310. In step 310, the index n is incremented by one, and then the procedure of step 204 is directed to step 304. In step 312, the core tensor Z is solved for as follows:

Z=D× ₁ U _(i) ^(T) × ₂ U ₂ ^(T) . . . ×_(n) . . . ×_(N) U _(N) ^(T).

When the core tensor Z is selected, the procedure of step 204 is completed.

It should be noted that when D_((n)) is a non-square matrix, the computation of U_(n) in the singular value decomposition D_((n))=U_(n)ΣV_(n) ^(T) can be performed, depending on which dimension of D_((n)) is smaller, by decomposing either D_((n))D_((n)) ^(T)=U_(n)Σ²U_(n) ^(T) and then computing v_(n) ^(T)=Σ⁺U_(n) ^(T)D_((n)), or by decomposing D_((n)) ^(T)D_((n))=v_(n)Σ²v_(n) ^(T) and then computing U_(n)=D_((n))V_(n)Σ⁺.

FIG. 4 illustrates the details of the individual generation procedure of step 208, which synthesizes the remaining actions, which were never before seen, for a new subject. The remaining actions are generated given the new motion data tensor D_(p,a) of the new subject performing action a, which includes at least one action. The individual generation procedure of step 208 solves for the signature p of the new individual in the equation D_(p,a)=B_(a)×₁p^(T), where B_(a)=Z×₂a_(a) ^(T)×₃J . It should be noted that new data tensor D_(p,a) is a 1×1×T tensor. In particular, step 402 of this procedure flattens the new data tensor D_(p,a) in the people (or subject) mode, yielding a row vector d_(a) ^(T). By flattening this new data tensor in the subject mode, the matrix D_(p,a(subject)) is generated, and in particular a row vector which we can denote as d_(a) ^(T) is produced. Therefore, in terms of the flattened tensors, the equation D_(p,a)=B_(a)×₁p^(T) described above can be written as d_(a) ^(T)=p^(T)B_(a(subject)) or d_(a)=B_(a(people)) ^(T)p. Once the tensor is flattened, the process advances to step 404, in which it is determined if the subject is observed performing a single action. If the subject is observed performing a single action, the procedure of step 208 is forwarded to step 406. If the individual is observed performing at least two actions, the procedure of step 208 advances to step 408. In step 406, the motion signature for the individual given a single observed action is computed. The motion signature for the individual can be defined as p^(T)=d_(a) ^(T)B_(a(people)) ⁻¹. When the motion signature for the individual is computed, the procedure of step 208 is completed. Also in step 408, the motion signature for the individual given at least two observed actions is determined. If several different actions d_(a,k) are observed, the motion signature can be computed as follows:

$p_{t} = {{\left\lbrack \mspace{11mu} {\ldots \mspace{14mu} d_{ak}^{T}\mspace{14mu} \ldots}\mspace{14mu} \right\rbrack \begin{bmatrix} \vdots \\ B_{{ak}{({people})}}^{- 1} \\ \vdots \end{bmatrix}}.}$

In step 410, the procedure of step 208 synthesizes a complete set of motions for the subject or individual. The complete set of motions for the new subject can be synthesized as follows:

D _(p) =B× ₁ p ^(T),

where B is defined as B=Z×₂A×₃J, as described above. When the motion signature for the individual is computed, the process 208 exits.

FIG. 5 illustrates details of the action generation procedure of step 212, which synthesizes an observed new action that has never before been seen for the remainder of the subjects represented in the subject matrix P. The observed action for the remainder of the subjects represented in the subject matrix P is generated given the new motion data tensor D_(p,a) of at least one subject performing the new action a.

In particular, step 501 of this procedure flattens the new data tensor D_(p,a) in the action mode, yielding a row vector d_(p) ^(T). By flattening this new data tensor in the action mode, the matrix D_(p,a(action)) is generated, and in particular a row vector which we can denote as d_(p) ^(T) is produced. Therefore, in terms of the flattened tensors, the equation D_(p,a)=C_(p)×₂a^(T) described above can be written as d_(p) ^(T)=a^(T)C_(p(actions)) or d_(p)=C_(p(actions)) ^(T). Once the tensor is flattened, this procedure determines as to whether the new motion data tensor D_(p,a) represents one subject performing the new action in step 502. If the new motion data tensor D_(p,a) represents one subject performing the new action, the procedure of step 212 advances to step 504. If the new motion data tensor D_(p,a) represents more than one individual performing the new action, the procedure of step 212 is forwarded to step 506. In step 504, the associated action parameters are determined based on the new motion data tensor D_(p,a), which represents one subject performing the new action. If a known subject, e.g., a person who is already recorded in the motion database, performs a new type of action d_(p), it is possible to compute the associated action parameters a^(T)=d_(p) ^(T)C⁻¹ _(p(actions)). When the p(actions) associated action parameters are computed, the procedure of step 212 is directed to step 508.

In step 506, the associated action parameters are computed based on the new motion data tensor D_(p,a), which represents more than one subject performing the new action. If several different subjects are observed performing the same new action d_(pk), the action parameters are computed as follows:

$a^{T} = {{\left\lbrack \mspace{11mu} {\ldots \mspace{14mu} d_{Pk}^{T}\mspace{14mu} \ldots}\mspace{14mu} \right\rbrack \begin{bmatrix} \vdots \\ C_{{Pk}{({actions})}}^{- 1} \\ \vdots \end{bmatrix}}.}$

When the associated action parameters are computed, the process 212 advances to step 508, in which the new action are obtained for the remainder of the subjects represented in the subject matrix P. The new action for all the subjects in the database can be synthesized as follows: D_(a)=C×₂a^(T), where C is given as C=Z×₁P×₃J, supra. When the new action is synthesized, the procedure of step 212 is completed.

FIG. 6 illustrates an individual recognition procedure of step 216 for recognizing an unidentified subject performing a known action. Multilinear analysis, can provide basis tensors that map certain observed motions into the space of subject parameters (thereby enabling the recognition of people from motion data) or the space action parameters (thereby enabling the recognition of action from motion data). The individual recognition process 216 begins at step 602, in which the signature p of an unknown subject performing a known action is computed. The new motion vector d of a known action a can be mapped into the subject signature space, by computing the signature p=B^(−T) _(a(people))d. Once the signature is computed, the process 216 advances to step 604, in which an index variable n and a variable match are initialized. For example, the index variable n can be initialized to one (1) and the variable match may be initialized to negative one (−1). Once these variables are initialized, step 606 is performed in which, the signature p is compared to a subject signature p_(n). This signature is compared against each of the person signatures p_(n) in P. Then the magnitude of the difference between the signature p and the signature p_(n), i.e. ∥p-31 p_(n)∥ is determined.

Thereafter, in step 608, it is determined whether a process-computed magnitude of the difference between the signature p and the signature p_(n) is smaller than any magnitude computed up to this point. If the magnitude of the difference between the signature p and the signature p_(n) is smaller than any difference computed up to this point, the process 216 advances to step 610. Otherwise, the process 216 is forwarded to step 612. In step 610, the variable match is set to be equal to the index n. The variable match generally signifies the index of the recognized subject, such that the signature p most closely matches the signature p_(match).

Then, in step 612, it is determined if the index n is equal to G. If that is the case, the procedure of step 216 advances to step 616, otherwise the procedure of step 216 is forwarded to step 614. In step 614, the index n is incremented by one (1), and the procedure is returned to step 606, such that each of the subjects in the subject matrix P from 1 to G is subjected to the comparison. Finally, in step 616, the signature p_(match) is identified as the signature that most closely approximates the signature p. In a preferred embodiment of the present invention, the variable match is an indexed array, which records the indices of multiple signatures that most closely match the signature p. Once the signature p_(match) is identified, the procedure of step 216 is completed.

FIG. 7 illustrates the details of an action recognition procedure of step 220 for recognizing an unknown action being performed by a known subject. Generally, a multilinear analysis yields basis tensors that map the observed motions into the space of action parameters, thus enabling the recognition of actions from the motion data. In particular, step 702 computes the vector a of a known individual performing an unknown action. The new motion data vector d of a known person p can be mapped into the action parameter space by computing the vector a=C^(−T) _(p(actions))d . When the vector a is determined, the procedure of step 220 advances to step 704, in which an index variable m and a variable match are initialized. The index variable m can be initialized to one (1), and the variable match may be initialized to negative one (−1). Once these variables are initialized, the process 220 is forwarded to step 706, in which the vector a is compared to an action parameter vector a_(m). In particular, the vector a is compared against each of the action parameter vectors a_(m) in A, in turn, as the index m is incremented. The magnitude of the difference between the vector a and the action parameter vector a_(m), i.e. ∥a−a_(m)∥, is also determined.

In step 708, the procedure of step 220 determines whether process computed magnitude of the difference between the vector a and the action parameter vector a_(m) is smaller than any difference computed up to this point. If the magnitude of the difference between the vector a and the action parameter vector a_(m) is smaller than any difference computed up to this point, the procedure of step 220 advances to step 710. Otherwise, the procedure of step 220 is forwarded to step 712. In step 710, the procedure of step 220 sets the variable match is set to be equal to the index m. The variable match generally signifies the index of the recognized action, such that the vector a most closely matches the action parameter vector a_(match).

Then, in step 712, it is determined if the index in is equal to M. If that is the case, the procedure of step 220 advances to step 716, otherwise the procedure is forwarded to step 714. Step 714, indicates that the index m is incremented by one (1), and the procedure advances to step 706, such that the index m increments through each of the actions in the action matrix A from 1 to M. In step 714, the action parameter vector a_(match) is identified as the signature that most closely approximates the vector a. In a preferred embodiment of the present invention, the variable match can be an indexed array, which records the indices of multiple actions that most closely match the vector a. Once the action parameter vector a_(match) is identified, the procedure of step 220 is completed.

B. Facial Signatures Using A Tensor Representation of A Corpus of Data

FIG. 8 illustrates a flow diagram of an exemplary embodiment of a process implementing a multilinear data analysis application 800 according to the present invention. As described above, the multilinear data analysis application 800 may be configured to recognize the unknown subject, unknown expression, unknown viewpoint and unknown, and synthesize a known illumination never before recorded for the subject, dimensionally reduce the amount of data describing illuminations, etc. The multilinear data analysis application 800 utilizes a corpus of facial data, which is collected using the data capturing system 112 from different subjects. The corpus of facial information can be stored in the database 108 of the server 102. This corpus of facial information may describe the illuminations, the views, the expressions, and the subjects captured in images made of pixels. The corpus of facial information is organized as a tensor D. The tensor D takes the form of a IR^(G×V×I×E×P) tensor, where G is the number of subjects, V is the number of viewpoints, I is the number of illuminations, E is the number of expressions, and P is the number of pixels. It should be understood that the corpus of motion information can also be organized as a matrix D or a vector d. For example, if the information is organized as a matrix D, the process 800 preferably remains the same, but the underlying tensor procedures could be converted to matrix procedure equivalents. It should also be noted that representing the data contained in the tensor D may integrate multiple indices into a singular matrix index. Likewise, if the information is organized as a vector d, the process 800 preferably remains the same, but the underlying tensor procedures could be converted to vector procedure equivalents. It should also be noted that representing the data contained in the tensor D may integrate multiple indices into a singular vector index.

In a preferred embodiment of the present invention, three expressions can be collected for each person: e.g., smile, neutral, and yawn. Each expression may be captured in four different illuminations, i.e. light positions, and three different viewpoints. The four different illuminations may be one light from the center, one light from the right, one light from the left, and two lights one from the right and one from the left. The three different viewpoints may be center, 34 degrees to the right, and 34 degrees to the left. In another preferred embodiment of the present invention, further similar expressions are collected for each person such that each expression is captured in four different illuminations and two different viewpoints. For example, the four different illuminations are one light from the center, one light from the right, one light from the left, and two lights one from the right and one from the left. The two different viewpoints are 17 degrees to the right, and 17 degrees to the left. In still another exemplary embodiment of the present invention, each expression is captured in three different illuminations and five different viewpoints. For example, the three different illuminations are one light from the center, one light from the right, and one light from the left. Also, the five different viewpoints are center, 17 degrees to the right, 17 degrees to the left, 34 degrees to the right, and 34 degrees to the left.

As shown in FIG. 8 step 802 provides that the multilinear data analysis application 800 collects facial information describing the illumination, viewpoint, expression, and subject. New facial data is collected describing the illumination of individual pixels of views of expressions of subjects. If each of the illuminations, each of the views, each of the expressions and individual are known, the data is integrated to the tensor D. Otherwise, the data cannot be integrated into the tensor D until those pieces of information are determined. The data describing an unknown illumination, view, expression or individual is organized as a new data vector d. The new data vector d describes an image having certain illumination, view, and expression data. Then in step 804, the multilinear data analysis application 800 solves for the core tensor Z. For example, this step can be an N-mode SVD procedure 304 as shown in FIG. 3 and described below in relation to FIG. 3. The N-mode SVD procedure 304 solves for the core tensor Z with N being equal to 5. When the procedure 804 or 304 computes the core tensor Z, the multilinear data analysis application 800 advances to step 806. Given the tensor D takes the form of a IR^(G×V×I×E×P) tensor, where G is the number of subjects, V is the number of viewpoints, I is the number of illuminations, E is the number of expressions, and P is the number of pixels. The N-mode SVD process 804 decomposed the tensor D as follows:

D=Z× ₁ U _(subjects)×₂ U _(views)×₃ U _(illum)×₄ U _(express)×₅ U _(pixels)

where the G×V×I×E×P core tensor Z governs the interaction between the factors represented in the 5 mode matrices: The G×G mode matrix U_(subjects) spans the space of subject parameters, the V×V mode matrix U_(views) spans the space of viewpoint . parameters, the I×I mode matrix U_(illum) spans the space of illumination parameters and the E×E mode matrix U_(express) spans the space of expression parameters. The P×P mode matrix U_(pixals) orthonormally spans the space of images.

The multilinear data analysis incorporates aspects of a linear principal component analysis (“PCA”) analysis. Each column of U_(subjects) is an “eigenimage”. These eigenimages are preferably identical to the conventional eigenfaces, since the eigenimages are computed by performing the SVD on the mode-5 flattened data tensor D so as to yield the matrix D_(subjects). One of the advantages of multilinear analysis is that the core tensor Z can transform the eigenimages in U_(pixels) into a set of eigenmodes, which represent the principal axes of variation across the various modes (subject, viewpoints, illuminations, expressions), and represent how the various factors interact with each other to create the facial images. This can be accomplished by generating the product Z×₅U_(pixels). In contrast, the PCA basis vectors or eigenimages represent only the principal axes of variation across images.

The facial image database can include V·I·E images for each subject which vary with viewpoint, illumination and expression. The PCA output represents each subject as a set of V·I·E vector-valued co-efficients, one from each image in which the subject appears.

Multilinear analysis allows each subject to be represented, regardless of viewpoint, illumination, and expression, with the same coefficient vector of dimension G relative to the bases comprising the G×V×I×E×P tensor

D=Z× ₂ U _(views)×₃ U _(illum)×₄ U _(express×) ₅ U _(pixels).

Each column in the tensor D is a basis matrix that comprises N eigenvectors. In any column, the first eigenvector depicts the average subject, and the remaining eigenvectors capture the variability across subjects for the particular combination of viewpoint, illumination and expression associated with that column. Each image is represented with a set of coefficient vectors representing the subject, view point, illumination and expression factors that generated the image. Multilinear decomposition allows the multilinear data analysis application 800 to construct different types of basis depending on the instruction received from the client interface application.

In particular step 814 of FIG. 8 provides that the multilinear data analysis application 800 determines whether the client interface application has instructed the multilinear data analysis application 800 to recognize an unknown subject who has been observed displaying a known expression as one of the population of observed known subjects. If the multilinear data analysis application 800 has received such instruction, the multilinear data analysis application 800 advances to an individual recognition procedure of step 816, shown in greater detail in FIG. 9 and described infra. When the individual recognition procedure of step 816 is completed as the multilinear data analysis application 800 advances to step 826. In step 818, the multilinear data analysis application 800 determines whether the client interface application has instructed the multilinear data analysis application 800 to recognize an unknown expression being displayed by a known subject as one of the expressions already observed as being performed by such known subject. If the multilinear data analysis application 800 has received such instruction, the multilinear data analysis application 800 advances to an expression recognition procedure of step 820, as shown in greater detail in FIG. 10 and described infra. When the expression recognition procedure of step 820 is completed, the multilinear data analysis application 800 is forwarded to step 826.

Thereafter, in step 822, the multilinear data analysis application 800 determines whether the client interface application has instructed the multilinear data analysis application 800 to dimensionally reduce the amount of data describing illuminations. If the multilinear data analysis application 800 has received such instruction, the multilinear data analysis application 800 advances to a data reduction procedure of step 824, as shown in greater detail in FIG. 11 and described infra. Once the data reduction procedure of step 824 is complete, the multilinear data analysis application 800 advances to step 826. Finally, in step 826, the multilinear data analysis application 800 determines whether a data set for a new subject should be collected or if the client interface application transmitted new instruction. If a data set for a new subject displaying an expression (e.g., a facial expression) is available, the multilinear data analysis application 800 advances to step 802. If the multilinear data analysis application 800 has received a new instruction from the client interface application, the multilinear data analysis application 800 advances to step 814.

FIG. 9 illustrates a flow diagram of the details of the individual recognition procedure of step 816 for recognizing an unidentified subject given an unknown facial image: the new data vector d. The multilinear data analysis preferably yields a basis tensor (as defined below) that maps all images of a subject to the same point in the subject parameter space, thus creating a many-to-one mapping. The individual recognition procedure of step 816 begins at step 902, in which the matrix U_(subjects) is extracted. The N-mode SVD procedure of step 804 (or step 304) decomposes the tensor D resulting in the expression D=Z×₁U_(subjects)×₂U_(views)×₃ U_(illum)×₄U_(express)×₅U_(pixels), and the matrix U_(subjects) is extracted from this expression. In particular, the matrix U_(subjects) contains row vectors c_(p) ^(T) of coefficients for each person p. Once the matrix U_(subjects) is extracted, the procedure of step 816 advances to step 904, in which the basis tensor B is generated. The basis tensor B is constructed according to B=Z×₂U_(views)×₃U_(illum)×₄U_(express)×₅U_(pixels). Upon the completion of the construction of the basis tensor B the procedure of step 816 advances to step 906 where this procedure initializes indexes v, i and e to one (1). At step 908, the individual recognition procedure of step 816 indexes into the basis tensor B to obtain a sub-tensor B_(v,i,e). This is performed for a particular viewpoint v, illumination i, and expression e to obtain the subtensor B_(v,i,e) having dimensions G×1×1×1×P.

Then, in step 910, the subtensor B_(v,i,e) is flattened along the subject mode. The subtensor B_(v,i,e) is flattened along the subject mode to obtain the G×P matrix B_(v,i,e(subject)). It should be noted that a specific training image d_(d) of subject p in viewpoint v, illumination i, and expression e can be written as d_(p,v,i,e)=B_(v,i,e(subject)C_(P); hence, c_(p)=B^(−T) _(v,i,e(subject))d_(p,v,i,e).

Then, in step 912, an index variable p and a variable match are initialized. For example, the index variable p is initialized to one (1), and the variable match is initialized to negative one (−1). Once these variables are initialized, the procedure of step 816 advances to step 914, in which. the projection operator B^(−T) _(v,i,e(subject)) is used to project the new data vector d into a set of candidate coefficient vectors. Given the new data vector d, the projection operator B^(−T) _(v,i,e(subject)) is used to project the new data vector d into a set of candidate coefficient vectors c_(v,i,e)=B^(−T) _(v,i,e(subject))d for every v, i, e combination. In step 916, each of the set of candidate coefficient vectors c_(v,i,e) is compared against the person-specific coefficient vectors c_(p). The comparison can be made according to the following equation:

μc_(v,i,e)−c_(p)∥.

In step 918, it is determined whether the set of candidate coefficient vectors c_(v,i,e) is the closest match to the subject-specific coefficient vectors c_(p) up to this point. The best matching vector c_(p) can be the one that yields the smallest value of ∥c_(v,i,e)−c_(p)∥ among all viewpoints, illuminations, and expressions. If the magnitude of the difference between the set of candidate coefficient vectors c_(v,i,e) and the subject-specific coefficient vectors c_(p) is smaller than any difference computed up to this point, the procedure of step 816 advances to step 920. Otherwise, the magnitude of the difference between the set of candidate coefficient vectors c_(v,i,e) and the procedure of step 816 is forwarded to step 922. Step 920 provides that the variable match is set to be equal to the index p. The variable match signifies the index of the most closely matched subject, such that the set of candidate coefficient vectors c_(v,i,e) most closely matches the subject-specific coefficient vectors c_(match).

Thereafter, in step 922, it is determined if the index p is equal to G. If that is the case, the procedure of step 816 sets the index p is set equal to one (1) and advances to step 928; otherwise, the procedure of step 816 advances to step 924. In step 924, the index p is incremented by one (1), and the procedure of step 816 advances to step 914, such that the procedure tests each of the subjects in the subject matrix U_(subject) from 1 to G.

In step 928, it is determined if the index e is equal to E. If that is the case, the procedure of step 816 sets the index e equal to one (1) and advances to step 930; otherwise, the procedure of step 816 advances to step 934. In step 934, the index e is incremented by one (1), and the procedure of step 816 advances to step 908, such that the procedure tests each of the subjects in the subject matrix U_(express) from 1 to E.

In step 930, it is determined if the index i is equal to I. If that is the case, the procedure of step 816 sets the index i equal to one (1) and advances to step 932; otherwise, the procedure of step 816 advances to step 936. In step 936, the index i is incremented by one (1), and the procedure of step 816 advances to step 908, such that the procedure tests each of the subjects in the subject matrix U_(illum) from 1 to I.

In step 932, it is determined if the index v is equal to V. If that is the case, the procedure of step 816 advances to step 926; otherwise, the procedure of step 816 advances to step 938. In step 938, the index v is incremented by one (1), and the procedure of step 816 advances to step 908, such that the procedure tests each of the subjects in the subject matrix U_(views) from 1 to V. Finally, in step 926, the subject match can be identified as the subject portrayed in the new data vector d. In a preferred embodiment of the present invention, the variable match can be an indexed array, that records the indices of multiple subjects most closely matching the subjects portrayed in the new data vector d. Once the subject match is identified, the procedure of step 816 is completed.

FIG. 10 illustrates a flow diagram of the details of the expression recognition procedure of step 820 for recognizing an unidentified expression given an unknown facial image: the new data vector d. The expression recognition procedure of step 820 is largely the same as the subject recognition procedure of step 816. The expression recognition procedure of step 820 begins in step 1002, in which the matrix U_(express) is extracted, in a manner similar to that used to extract U_(subjects) in step 902. In particular, the matrix U_(express) contains row vectors c_(e) ^(T) of coefficients for each expression e. Once the matrix U_(express) is extracted, the procedure of step 820 advances to step 1004, in which the basis tensor B is generated. The basis tensor B is constructed according to B=Z×₂U_(views)×₃U_(illum)×₁U_(subjects)×₅U_(pixels). Upon the completion of the construction of the basis tensor B the procedure of step 820 advances to step 1006 where this procedure initializes indexes v, i and p to one (1). At step 1008, the expression recognition procedure of step 820 indexes into the basis tensor B to obtain a sub-tensor B_(p,v,i). This is performed for a particular subject p, viewpoint v and illumination i to obtain the subtensor B_(p,v,i) having dimensions 1×1×1×E×P.

Then, in step 1010, the subtensor B_(p,v,i) is flattened along the expression mode. The subtensor B_(p,v,i) is flattened along the expression mode to obtain the E×P matrix B_(p,v,i(express)). It should be noted that a specific training image d_(d) of subject p in viewpoint v, illumination i, and expression e can be written as d_(p,v,i,e)=B^(T) _(p,v,i (subject))c_(e); hence, c_(e)=B_(p,v,i(subject))d_(p,v,i,e).

Then, in step 1012, an index variable e and a variable match are initialized. For example, the index variable e is initialized to one (1), and the variable match is initialized to negative one (−1). Once these variables are initialized, the procedure of step 820 advances to step 1014, in which. the projection operator B^(−T) _(p,v,i(subject)) is used to project the new data vector d into a set of candidate coefficient vectors. Given the new data vector d, the projection operator B^(−T) _(p,v,i(subject)) is used to project the new data vector d into a set of candidate coefficient vectors c_(p,v,i)=B^(−T) _(p,v,i(subject))d for every p, v, i combination. In step 1016, each of the set of candidate coefficient vectors c_(p,v,i) is compared against the person-specific coefficient vectors c_(e).

The comparison can be made according to the following equation:

∥c_(p,v,i)−c_(e)∥.

In step 1018, it is determined whether the set of candidate coefficient vectors c_(p,v,i) is the closest match to the expression coefficient vectors c_(e) up to this point. The best matching vector c_(e) can be the one that yields the smallest value of ∥c_(p,v,i)−c_(e)∥ among all viewpoints, illuminations, and expressions. If the magnitude of the difference between the set of candidate coefficient vectors c_(p,v,i) and the expression coefficient vectors c_(e) is smaller than any difference computed up to this point, the procedure of step 820 advances to step 1020. Otherwise, the magnitude of the difference between the set of candidate coefficient vectors c_(p,v,i) and the procedure of step 820 is forwarded to step 1022. Step 1020 provides that the variable match is set to be equal to the index p. The variable match signifies the index of the most closely matched expression, such that the set of candidate coefficient vectors c_(p,v,i) most closely matches the expression coefficient vectors c_(match).

Thereafter, in step 1022, it is determined if the index e is equal to E. If that is the case, the procedure of step 820 sets the index p is set equal to one (1) and advances to step 1028; otherwise, the procedure of step 820 advances to step 1024. In step 1024, the index p is incremented by one (1), and the procedure of step 820 advances to step 1014, such that the procedure tests each of the expressions in the expression matrix U_(express) from 1 to E.

In step 1028, it is determined if the index p is equal to G. If that is the case, the procedure of step 820 sets the index e equal to one (1) and advances to step 1030; otherwise, the procedure of step 820 advances to step 1034. In step 1034, the index p is incremented by one (1), and the procedure of step 820 advances to step 1008, such that the procedure tests each of the subjects in the subject matrix U_(subject) from 1 to G.

In step 1030, it is determined if the index i is equal to I. If that is the case, the procedure of step 820 sets the index i equal to one (1) and advances to step 1032; otherwise, the procedure of step 820 advances to step 1036. In step 1036, the index i is incremented by one (1), and the procedure of step 820 advances to step 1008, such that the procedure tests each of the illuminations in the illumination matrix U_(illum) from 1 to I.

In step 1032, it is determined if the index v is equal to V. If that is the case, the procedure of step 820 advances to step 1026; otherwise, the procedure of step 820 advances to step 1038. In step 1038, the index v is incremented by one (1), and the procedure of step 820 advances to step 1008, such that the procedure tests each of the views in the view matrix U_(views) from 1 to V. Finally, in step 1026, the subject match can be identified as the subject portrayed in the new data vector d. In a preferred embodiment of the present invention, the variable match can be an indexed array, that records the indices of multiple subjects most closely matching the subjects portrayed in the new data vector d. Once the subject match is identified, the procedure of step 820 is completed.

FIG. 11 illustrates a flow diagram of the details for the data reduction procedure step 824 for dimensionally reduce the amount of data describing illuminations. This data reduction procedure step 824 reduces the amount of data by truncating the mode matrices resulting from the N-mode SVD procedure 304 or 804, where N=5. The truncation of the mode matrices yields an exemplary reduced-dimensionality approximation D′. The truncation of the mode matrices results in the approximation of the tensor D with reduced ranks R₁≦I₁,R₂≦I₂, . . . , R_(N)≦I_(N) that has a bounded error

${{{D - D^{\prime}}}^{2} \leq {{\sum\limits_{i_{1} = {R_{1} + 1}}^{I_{1}}\sigma_{i_{1}}^{2}} + \ldots + {\sum\limits_{i_{N} = {R_{N} + 1}}^{I_{N}}\sigma_{i_{N}}^{2}}}},$

where the smallest mode-n singular values that were discarded are defined as σ _(n) _(=R) _(n) ₊₁σ_(i) _(n) _(=R) _(n) ₊₂, . . . ,σ_(i) _(n) _(=I) _(n) . The R^(th) _(n) mode-n eigenvalue is the Frobenius norm of the subtensor Z_(i) ₁ _(, . . . ,i) _(n) _(=m, . . . ,i) _(N) . The subtensor Z_(i) ₁ _(, . . . ,i) _(n) _(=R) _(n) _(, . . . ,i) _(N) is extracted from the tensor Z by holding the n^(th) dimension fixed to i_(n)=R_(n) and varying all other dimensions. Once the index n is initialized, the procedure step 824 advances to step 1104.

In another exemplary dimensionality reduction procedure for use on the tensors is to compute for a tensor D a best rank−(R₁, R₂, . . . , R_(N)) approximation D′=Z′×₁U′₁×₂U′×₂ . . . ×_(N)U′_(N), with orthonormal I_(n)×R_(n) mode matrices U′_(n), for n=1, 2, . . . , N, which can minimize the least-squares error function ∥D−D′∥². For example, N can equal to five (5). The data reduction procedure step 824 begins in step 1102, where an index n is initialized to one (1).

In step 1104, the mode matrix U_(n) is truncated to R_(n) columns. All data in the mode matrix U_(n) beyond the R_(n) column can be removed from the matrix U_(n). After the matrix U_(n) is truncated, the procedure step 824 advances to step 1106, in which it is determined whether the index n is equal to N. If that is the case, the procedure step 824 advances to step 1110; otherwise, the procedure step 824 is forwarded to step 1108. In step 1108, the index n is incremented by one (1), and the procedure step 824 proceeds to step 1104. Then, in step 1110, the index n is initialized to one (1), and the procedure step 824 advances to step 1112, in which the tensor is calculated Ũ_(n) ^(k+1)=D×₂ ^(k) ^(T) ×₃U₃ ^(k) ^(T) . . . ×_(N)U_(N) ^(k) ^(T) . When the tensor U′_(n) ^(k+1) is calculated, the procedure step 824 advances to step 1114, in which the tensor U′_(n) ^(k+1) is mode-n flattened to obtain the matrix U′_(m(n)) ^(k+1). Then in step 1116, the matrix U′₁ ^(k+1) is computed as the I₁×R₁ matrix whose columns are the first R₁ columns of the left matrix of the SVD of U′₁₍₁₎ ^(k+1).

In step 1118, it is determined whether the index n is equal to N. If that is the case, the procedure step 824 advances to step 1122; otherwise the procedure step 824 advances to step 1120, in which the index n is incremented by one (1) and the procedure step 824 advances to step 1112. Then in step 1122, it is determined whether the mode matrices have converged. The mode matrices have converged if

U_(n)^(k + 1^(T))U_(n)^(k)² > (1−∈)R_(n), for  1 ≤ n ≤ N.

If the mode matrices have converged, the procedure step 824 advances to step 1124; otherwise the procedure step 824 advances to step 1110. In step 1124, the core tensor Z′ is computed. The converged mode matrices U′₁, U′₂ . . . , U′_(N) is used to compute the core tensor Z′=U′_(N)×_(n)U′_(N) ^(T) and D′=Z′×₁U′₁×₂U′₂ . . . ×_(N)U′_(N) as the rank-reduced approximation of the tensor D. Once the core tensor Z′ is computed, the procedure step 824 is completed.

C. Motion Signature Using A Matrix Representation of A Corpus of Data

FIG. 13 illustrates a flow diagram of an exemplary embodiment of a process implementing a multilinear data analysis application 1300 which is indicative of the multilinear data analysis application. As described above for the multilinear data analysis application, the process 1300 is configured to synthesize a known action never before recorded as being performed by the subject. In particular the multilinear data analysis application utilizes the corpus of motion data, which is collected using the data capturing system 112 from different subjects as described above in relation to FIG. 2. This corpus of motion information is stored in the database 108 of the server 102, and describes angles of the joints in the legs of at least one subject performing at least one action. The corpus of motion information can be organized as a matrix D and is preferably collected from different subjects as described above in relation to FIG. 2. It should be understood that the corpus of motion information can also be organized as a tensor D or a vector d. The multilinear data analysis application 1300 is similar to the multilinear data analysis application 200 of FIG. 2, except that the data utilized by the multilinear data analysis application 1300 takes is organized as the matrix D, not as the tensor D.

Turning to further particulars of FIG. 13, in step 1302, the process 1300 collects motion information or data on various subjects (e.g., people) performing different actions, e.g., new motion data. If the action and individual are known, the data can be integrated into the matrix D. If the action or individual are not known, such data would likely not be integrated into the matrix D until those pieces of information are determined. The data describing an unknown action or individual is organized as a new data matrix D_(p) or a new data vector d. The new data matrix D_(p) can include more than one new data vector d. Each new data vector d_(p,a) of the new data matrix D_(p) describes the motion of subject p performing action a. With the knowledge of motion sequences of several subjects, the matrix D can take the form of a nt×m matrix, where n is the number of subjects, t is the number of joint angle time samples, and m is the number of motion classes. The first column of the matrix D stacks the mean walk of every subject, the second column stacks the mean ascending motion and the third stacks the mean stair descent, as follows:

$D = \begin{bmatrix} D_{I} \\ \vdots \\ D_{i} \\ \vdots \\ D_{n} \end{bmatrix}$ $D_{i} = \begin{bmatrix} \rightarrow & \rightarrow & \rightarrow \\ {walk}_{\; i} & {ascend}_{\; i} & {descend}_{\; i} \end{bmatrix}$

The columns of the matrix D_(i) are the average walk, ascend and descend of stairs of the i^(th) subject. Each motion is defined as the angles by every joint over time.

At step 1304, the process 1300 decomposes the matrix D into a core matrix Z, a subject matrix P, and an action matrix A. The core matrix Z can be used for defining the inter-relationships between a subjects matrix P and an action matrix A. This step represents a singular value decomposition (“SVD”) process 1304, shown in FIG. 14, and described in further detail herein. The SVD procedure of step 1304 is an orthonormal procedure that solves for the core matrix Z, the subject matrix P, and the action matrix A, which minimizes

E=∥D−(Z ^(VT) P ^(T))^(VT) A ^(T)∥+λ₁ ∥P ^(T) P−I∥+λ ₂ ∥A ^(T) A−I∥,

where I is the identity matrix. When this procedure of step 1304 determines the core matrix Z, the process 1300 advances to step 1305.

In step 1305, the process 1300 analyzes the data collected in the step 1302. The SVD procedure of step 1304 decomposes the matrix D into the product of a core matrix Z, and two orthogonal matrices as follows:

$\begin{matrix} {D = {\left( {Z^{VT}P^{T}} \right)^{VT}A^{T}}} \\ {{= {SA}^{T}},} \end{matrix}$

where the VT-operator is a matrix transpose T followed by a “vec” operator that creates a vector by stacking the columns of the matrix. The subject matrix P=[p₁. . . p_(n) . . . p_(G)]^(T), whose row vectors p_(i) are person specific, encodes the invariancies across actions for each person. Thus, the subject matrix P contains the subject or human motion signatures p_(i). The action matrix

$A = \begin{matrix} \left\lbrack \vdots \right. & \vdots & \left. \vdots \right\rbrack^{T} \\  &  &  \\ a_{walk} & a_{ascend} & a_{descend} \end{matrix}$

whose row vectors a_(c), contain the coefficients for the different action classes c, encodes the invariancies across subjects for each action. The core matrix Z=[Z₁ . . . Z_(i) . . . Z_(n)]^(T) represents the basis motions which are independent of people and of actions. It governs the relationship between the orthonormal matrices P and A. A matrix

S=(Z ^(VT) P ^(T))^(VT) =[S ₁ . . . S _(i) . . . S _(n)]^(T)

is composed of person-specific signature matrices S.

In step 1306, the process 1300 determines whether it has been instructed by the client interface application to synthesize new data describing at least one known action that was never before recorded as being performed by a subject. If the process 1300 has received such instruction, step 1308 is executed to perform advances to an individual generation procedure, as shown in further detail in FIG. 15 and described herein. When the individual generation procedure of step 1308 is complete, the process 1300 advances to step 1326. Then in step 1326, the process 1300 determines whether a data set for a new subject should be integrated into the matrix D or if the client interface application has transmitted a new instruction. In particular, if the data set for a new subject performing the action is available, the process 1300 advances to step 1302. Otherwise, the process 1300 received the new instruction from the client interface application, so the process 1300 continues to step 1306.

As shown in FIG. 14, the procedure of step 1304 begins in step 1402 by computing the matrix P by solving D=(Z^(VT)P^(T))^(VT)A^(T). The process then calculates (DA)^(VT)=Z^(VT)P^(T). The procedure performs an SVD procedure on the left hand side resulting in USV^(T)=Z^(VT)P^(T). The matrix V is then truncated to the first r-columns of the matrix V. The procedure of step 1304 then solves for the action matrix A in step 1404 by calculating D^(VT)=(ZA^(T))^(VT)P^(T). Once this is calculated, the procedure calculates (D^(VT)P)^(VT)=ZA^(T). The procedure performs SVD on the left hand side resulting in USV^(T)=ZA^(T). The matrix A is then truncated to the first r-columns of the matrix V. In step 1406, the procedure of step 1304 obtains the core matrix Z by Z=(D^(VT)P)^(VT)A, where the matrix P and the matrix A are orthonormal. It should be understood that by setting the matrix A and the matrix P to the first r-columns of the matrix V, effectively accomplishing dimensional reduction.

FIG. 15 illustrates the details of the individual generation procedure of step 1308, which synthesizes the remaining actions, which were never before seen, for an new individual. The remaining actions are generated given new motion data D_(new) of the new subject performing an action. The new signature model of the new subject is the matrix

$D_{new}\left\{ {\left\lbrack ? \right.\left.  \right\rbrack \underset{\underset{Snew}{}}{\left\lbrack \left. ? \right\rbrack \right.}{A^{T}.}} \right.$

Only a portion of the action classes c are represented the matrix D_(new). The linear combination of known signatures is:

$S_{new} = {\underset{\underset{W}{}}{\begin{bmatrix} W_{I} & \ldots & W_{i} & \ldots & W_{n} \end{bmatrix}}\underset{\underset{S}{}}{\begin{bmatrix} S_{I} \\ \vdots \\ S_{i} \\ \vdots \\ S_{n} \end{bmatrix}}}$

where W is a weight matrix. The individual generation procedure of step 1308 solves for the weight matrix W of the new subject using iterative gradient descent of the error function

E=∥D _(new) −WSA _(inc) ^(T)∥,

where A_(inc) ^(T) has only columns corresponding to the motion examples available in the matrix D_(new). In particular, step 1502 of this procedure initializes an index t to one (1). In step 1504, the procedure of step 1308 obtains the matrix Q by calculating Q=SA_(inc) ^(T). Once this procedure obtains the matrix Q, step 1506 of the procedure of step 1308 calculates the matrix W(t+1) in the following manner: W(t+1)=W(t)+γ(D_(new)−WQ)Q^(T). The step 1508 then calculates S_(new)(t+1) by calculating S_(new)(t+1)=W(t+1)S, then this procedure advances to step 1510.

In step 1510, it is determined whether the error function E has converged. If the error function E has not converged, the procedure of step 1308 continues to step 1512, where the index t is incremented by one (1) and this procedure advances to step 1504. If the error function E has converged, this procedure advances to step 1514. In step 1514 the procedure of step 1308 synthesizes new data from one of the action parameters c. For example, if the action parameter c represents the action of walking. The new data for walking is synthesized by multiplying the newly extracted signature matrix S_(new) and the action parameters for walking, a_(walk), as follows:

{right arrow over (walk)}_(new) S _(new) {right arrow over (a)} _(walk).

Once the new data is synthesized, the procedure of step 1308 is complete and it exits.

While the invention has been described in connecting with preferred embodiments, it will be understood by those of ordinary skill in the art that other variations and modifications of the preferred embodiments described above may be made without departing from the scope of the invention. Other embodiments will be apparent to those of ordinary skill in the art from a consideration of the specification or practice of the invention disclosed herein. It is intended that the specification and the described examples are considered as exemplary only, with the true scope and spirit of the invention indicated by the following claims. 

1-136. (canceled)
 137. A computer-accessible medium which includes a computer program thereon for image-based recognition wherein, when the computer program is executed by a processing arrangement, the processing arrangement is configured to: receive image data; generate an orthonormal decomposition of the image data related to an invariance of motions associated with the image data; and recognize at least one of at least one subject, an action, an expression, an illumination or a viewpoint based on the orthonormal decomposition.
 138. The computer-accessible medium of claim 137, wherein the motions are associated with a plurality of subjects.
 139. The computer-accessible medium of claim 137, wherein the orthonormal decomposition comprises an n-mode singular value decomposition (“SVD”).
 140. A computer-accessible medium which includes a computer program thereon for image generation wherein, when the computer program is executed by a processing arrangement, the processing arrangement is configured to: receive first image data; generate motion signature data using an orthonormal decomposition of the first image data relating to an invariance of motions associated with the first image data; and generate second image data related to at least one of an action, an expression, an illumination or a viewpoint based on the motion signature data.
 141. The computer-accessible medium of claim 140, wherein the at least one of the action, the expression, the illumination or the viewpoint is automatically synthesized.
 142. The computer-accessible medium of claim 140, wherein the orthonormal decomposition of the image data relates to an invariance of motions across a plurality of subjects.
 143. The computer-accessible medium of claim 140, wherein the orthonormal decomposition is an n-mode singular value decomposition (“SVD”).
 144. A system for image-based recognition, comprising: a processing arrangement which, when executed, is configured to: receive image data; generate an orthonormal decomposition of the image data related to an invariance of motions associated with the image data; and recognize at least one of at least one subject, an action, an expression, an illumination or a viewpoint based on the orthonormal decomposition.
 145. The system of claim 144, wherein the motions are associated with a plurality of subjects.
 146. The system of claim 144, wherein the orthonormal decomposition comprises an n-mode singular value decomposition (“SVD”).
 147. A system for image generation, comprising: a processing arrangement which, when executed, is configured to: receive first image data; generate motion signature data using an orthonormal decomposition of the first image data relating to an invariance of motions associated with the first image data; and generate second image data related to at least one of an action, an expression, an illumination or a viewpoint based on the motion signature data.
 148. The system of claim 147, wherein the at least one of the action, the expression, the illumination or the viewpoint is automatically synthesized.
 149. The system of claim 147, wherein the motions are associated with a plurality of subjects.
 150. The system of claim 147, wherein the orthonormal decomposition comprises an n-mode singular value decomposition (“SVD”).
 151. A method for image-based recognition, comprising: receiving image data; generating an orthonormal decomposition of the image data related to an invariance of motions associated with the image data; and using a processing arrangement, recognizing at least one of at least one subject, an action, an expression, an illumination or a viewpoint based on the orthonormal decomposition.
 152. The method of claim 151, wherein the motions are associated with a plurality of subjects.
 153. The method of claim 151, wherein the orthonormal decomposition comprises an n-mode singular value decomposition (“SVD”).
 154. A method for image generation, comprising: receiving first image data; generating motion signature data using an orthonormal decomposition of the first image data relating to an invariance of motions associated with the first image data; and using a processing arrangement, generating second image data related to at least one of an action, an expression, an illumination or a viewpoint based on the motion signature data, wherein the at least one of the action, the expression, the illumination or the viewpoint is synthesized.
 155. The method of claim 154, wherein the motions are associated with a plurality of subjects.
 156. The method of claim 154, wherein the orthonormal decomposition comprises an n-mode singular value decomposition (“SVD”). 